Physics 1161: Lecture 30 Models of the Atom Sections 31-1 – 31-4 Bohr model works, approximately Hydrogen-like energy levels (relative to a free electron that wanders off): mk 2 e4 Z 2 13.6 Z 2 En eV where h / 2 2 2 2 2 n n Z2 En 13.6 2 eV Energy of a Bohr orbit n Typical hydrogen-like radius (1 electron, Z protons): 2 2 h 1 n 2 rn 0.0529 nm n 2 2 mke Z rn 0.0529 nm n2 Radius of a Bohr orbit Preflight 30.1 h 2 1 n2 n2 rn ( ) (0.0529nm) 2 2 mke Z Z Bohr radius If the electron in the hydrogen atom was 207 times heavier (a muon), the Bohr radius would be 1) 207 Times Larger 19 % 2) Same Size 53 % 3) 207 Times Smaller 28 % (Z =1 for hydrogen) Preflight 30.1 h 2 1 n2 n2 rn ( ) (0.0529nm) 2 2 mke Z Z Bohr radius If the electron in the hydrogen atom was 207 times heavier (a muon), the Bohr radius would be 1) 207 Times Larger h 2 1 Bohr Radius ( ) 2 mke2 2) Same Size 3) 207 Times Smaller This “m” is electron mass, not proton mass! Preflight 30.2 A single electron is orbiting around a nucleus with charge +3. What is its ground state (n=1) energy? (Recall for charge +1, E= -13.6 eV) 1) 2) 3) E = 9 (-13.6 eV) E = 3 (-13.6 eV) E = 1 (-13.6 eV) 32/1 = 9 Z2 E n 13.6eV 2 n Note: This is LOWER energy since negative! Transitions + Energy Conservation • Each orbit has a specific energy: En= -13.6 Z2/n2 • Photon emitted when electron jumps from high energy to low energy orbit. Photon absorbed when electron jumps from low energy to high energy: | E1 – E2 | = h f = h c / l http://www.colorado.edu/physics/2000/quantumzone/bohr2.html Line Spectra elements emit a discrete set of wavelengths which show up as lines in a diffraction grating. Preflight 30.3 Electron A falls from energy level n=2 to energy level n=1 (ground state), causing a photon to be emitted. Electron B falls from energy level n=3 to energy level n=1 (ground state), causing a photon to be emitted. n=3 Which photon has more energy? • Photon A 22 % • Photon B 78 % n=2 A B n=1 Preflight 30.3 Electron A falls from energy level n=2 to energy level n=1 (ground state), causing a photon to be emitted. Electron B falls from energy level n=3 to energy level n=1 (ground state), causing a photon to be emitted. n=3 Which photon has more energy? n=2 • Photon A • Photon B A B n=1 Spectral Line Wavelengths Calculate the wavelength of photon emitted when an electron in the hydrogen atom drops from the n=2 state to the ground state (n=1). hf E2 E1 3.4eV (13.6eV) 10.2eV n=3 E2= -3.4 eV n=2 Ephoton hc l hc 1240 l 124nm 10.2eV 10.2 E1= -13.6 eV n=1 Compare the wavelength of a photon produced from a transition from n=3 to n=2 with that of a photon produced from a transition n=2 to n=1. 1. l32 < l21 2. l32 = l21 3. l32 > l21 n=3 n=2 33% 33% 33% n=1 1 2 3 Compare the wavelength of a photon produced from a transition from n=3 to n=2 with that of a photon produced from a transition n=2 to n=1. 1. l32 < l21 2. l32 = l21 3. l32 > l21 n=3 n=2 43% 37% n=1 20% E32 < E21 so l32 > l21 1 2 3 Preflight 30.4 The electrons in a large group of hydrogen atoms are excited to the n=3 level. How many spectral lines will be produced? (1) 11 % (2) 13 % (3) (4) 9% (5) (6) 2% n=3 n=2 n=1 57% 9% Preflights 30.6, 30.8 So what keeps the electron from “sticking” to the nucleus? Centripetal Acceleration Pauli Exclusion Principle 34 % 32% Heisenberg Uncertainty Principle 34 % To be consistent with the Heisenberg Uncertainty Principle, which of these properties can not be quantized (have the exact value known)? (more than one answer can be correct) 38 % Electron Orbital Radius 57 % Electron Energy 43 % Electron Velocity 43 % Electron Angular Momentum Would know location Would know momentum Quantum Mechanics • Predicts available energy states agreeing with Bohr. • Don’t have definite electron position, only a probability function. • Orbitals can have 0 angular momentum! • Each electron state labeled by 4 numbers: n = principal quantum number (1, 2, 3, …) l = angular momentum (0, 1, 2, … n-1) ml = component of l (-l < ml < l) Quantum ms = spin (-½ , +½) Numbers Summary • Bohr’s Model gives accurate values for electron energy levels... • But Quantum Mechanics is needed to describe electrons in atom. • Electrons jump between states by emitting or absorbing photons of the appropriate energy. • Each state has specific energy and is labeled by 4 quantum numbers (next time). JAVA Links • Bohr Atom • Debroglie Atom • Schroedinger Atom Bohr’s Model • Mini Universe • Coulomb attraction produces centripetal acceleration. – This gives energy for each allowed radius. • Spectra tells you which radii orbits are allowed. – Fits show this is equivalent to constraining angular momentum L = mvr = n h Bohr’s Derivation 1 Circular motion mv 2 kZe 2 r r2 2 1 kZe mv 2 2 2r 1 2 kZe 2 kZe 2 E mv 2 r 2r Total energy Quantization of angular momentum: h (mvr )n mv n rn n 2 h vn n 2mrn Bohr’s Derivation 2 Use h vn n 2mrn in 2 kZe mv n2 rn 2 h 1 n rn n 2 ( )2 (0.0529nm ) 2 2 mkZe Z “Bohr radius” Substitute for rn in kZe 2 En 2rn Z2 E n 13.6eV 2 n Note: rn has Z En has Z2 Quantum Numbers Each electron in an atom is labeled by 4 #’s n = Principal Quantum Number (1, 2, 3, …) • Determines energy ℓ = Orbital Quantum Number (0, 1, 2, … n-1) • Determines angular momentum • L h ( 1) 2 mℓ = Magnetic Quantum Number (ℓ , … 0, … -ℓ ) • Component of ℓ • ms = Spin Quantum Number (+½ , -½) • “Up Spin” or “Down Spin” h Lz m 2 Nomenclature “Shells” “Subshells” n=1 is “K shell” ℓ =0 is “s state” n=2 is “L shell” ℓ =1 is “p state” n=3 is “M shell” ℓ =2 is “d state” n=4 is “N shell” ℓ =3 is “f state” n=5 is “O shell” ℓ =4 is “g state” 1 electron in ground state of Hydrogen: n=1, ℓ =0 is denoted as: 1s1 n=1 ℓ =0 1 electron Quantum Numbers How many unique electron states exist with n=2? ℓ =0: 2s2 mℓ = 0 : ms = ½ , -½ ℓ =1: 2 states 2p6 mℓ = +1: ms = ½ , -½ mℓ = 0: ms = ½ , -½ mℓ = -1: ms = ½ , -½ 2 states 2 states 2 states There are a total of 8 states with n=2 How many unique electron states exist with n=5 and ml = +3? 1. 2. 3. 4. 2 3 4 5 0% 1 0% 2 0% 3 0% 4 How many unique electron states exist with n=5 and ml = +3? 1. 2 2. 3 3. 4 4. 5 ℓ =3: ℓ ℓ ℓ = 0 : mℓ = 0 ℓ = 1 : mℓ = -1, 0, +1 ℓ = 2 : mℓ = -2, -1, 0, +1, +2 Only = 3 and ℓ = 4 have mℓ = +3 mℓ = -3, -2, -1, 0, +1, +2, +3 ms = ½ , -½ 2 states = 4 : mℓ = -4, -3, -2, -1, 0, +1, +2, +3, +4 ms = ½ , -½ 2 states There are a total of 4 states with n=5, mℓ = +3 0% 1 0% 2 0% 3 0% 4 Pauli Exclusion Principle In an atom with many electrons only one electron is allowed in each quantum state (n, ℓ,mℓ,ms). This explains the periodic table! Preflight 31.2 What is the maximum number of electrons that can exist in the 5g (n=5, ℓ = 4) subshell of an atom? Preflight 31.2 What is the maximum number of electrons that can exist in the 5g (n=5, ℓ = 4) subshell of an atom? mℓ = -4 : ms = ½ , -½ 2 states mℓ = -3 : ms = ½ , -½ 2 states mℓ = -2 : ms = ½ , -½ 2 states mℓ = -1 : ms = ½ , -½ 2 states mℓ = 0 : ms = ½ , -½ 2 states mℓ = +1: ms = ½ , -½ 2 states mℓ = +2: ms = ½ , -½ 2 states mℓ= +3: ms = ½ , -½ 2 states mℓ = +4: ms = ½ , -½ 2 states 18 states Electron Configurations Atom Configuration H 1s1 He 1s2 Li 1s22s1 Be 1s22s2 B 1s22s22p1 1s shell filled (n=1 shell filled noble gas) 2s shell filled etc Ne 1s22s22p6 s shells hold up to 2 electrons 2p shell filled (n=2 shell filled noble gas) p shells hold up to 6 electrons Sequence of Shells Sequence of shells: 1s,2s,2p,3s,3p,4s,3d,4p….. 4s electrons get closer to nucleus than 3d 19 20 21 22 23 24 25 26 27 28 29 30 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn 4s 3d 4p In 3d shell we are putting electrons into ℓ = 2; all atoms in middle are strongly magnetic. Angular momentum Loop of current Large magnetic moment Sodium Na 1s22s22p6 3s1 Single outer electron Neon - like core Many spectral lines of Na are outer electron making transitions Yellow line of Na flame test is 3p 3s www.WebElements.com Summary • Each electron state labeled by 4 numbers: n = principal quantum number (1, 2, 3, …) ℓ = angular momentum (0, 1, 2, … n-1) mℓ = component of ℓ (-ℓ < mℓ < ℓ) ms = spin (-½ , +½) • Pauli Exclusion Principle explains periodic table • Shells fill in order of lowest energy.